10.1 Lines and Segments That Intersect Circles EEssentialssential QQuestionuestion What are the defi nitions of the lines and segments that intersect a circle? Lines and Line Segments That Intersect Circles Work with a partner. The drawing at tangent the right shows fi ve lines or segments that intersect a circle. Use the relationships shown to write a defi nition for each type of line or segment. Then use the Internet or some other resource to verify your defi nitions. chord Chord: radius Secant: diameter Tangent: secant Radius: Diameter: Using String to Draw a Circle Work with a partner. Use two pencils, a piece of string, and a piece of paper. a. Tie the two ends of the piece of string loosely around the two pencils. b. Anchor one pencil on the paper at the center of the circle. Use the other pencil to draw a circle around the anchor point while using slight pressure to keep the string taut. Do not let the string wind around either pencil. REASONING ABSTRACTLY To be profi cient in math, you need to know and fl exibly use different properties of operations and objects. c. Explain how the distance between the two pencil points as you draw the circle is related to two of the lines or line segments you defi ned in Exploration 1. CCommunicateommunicate YourYour AnswerAnswer 3. What are the defi nitions of the lines and segments that intersect a circle? 4. Of the fi ve types of lines and segments in Exploration 1, which one is a subset of another? Explain. 5. Explain how to draw a circle with a diameter of 8 inches. Section 10.1 Lines and Segments That Intersect Circles 529 hhs_geo_pe_1001.indds_geo_pe_1001.indd 529529 11/19/15/19/15 22:31:31 PMPM 10.1 Lesson WWhathat YYouou WWillill LLearnearn Identify special segments and lines. Draw and identify common tangents. Core VocabularyVocabulary Use properties of tangents. circle, p. 530 center, p. 530 Identifying Special Segments and Lines radius, p. 530 chord, p. 530 A circle is the set of all points in a plane that are equidistant from a given point called the center of the circle. A circle with center P diameter, p. 530 P is called “circle P” and can be written as ⊙P. secant, p. 530 tangent, p. 530 point of tangency, p. 530 circle P, or ⊙P tangent circles, p. 531 concentric circles, p. 531 CCoreore CConceptoncept common tangent, p. 531 Lines and Segments That Intersect Circles A segment whose endpoints are the center and chord any point on a circle is a radius. radius center A chord is a segment whose endpoints are on diameter a circle. A diameter is a chord that contains the READING center of the circle. The words “radius” and A secant is a line that intersects a circle in “diameter” refer to lengths two points. as well as segments. For a secant A tangent is a line in the plane of a circle that given circle, think of a radius intersects the circle in exactly one point, the point of and a diameter as segments point of tangency. The tangent ray ៮AB៮៮⃗ and tangency and the radius and the — the tangent segment AB are also called tangents. tangent B A diameter as lengths. Identifying Special Segments and Lines Tell whether the line, ray, or segment is best described as D a radius, chord, diameter, secant, or tangent of ⊙C. — — C a. AC b. AB A B c. DE៮៮៮⃗ d. ⃖AE៮៮⃗ E G STUDY TIP SOLUTION In this book, assume that all — a. AC is a radius because C is the center and A is a point on the circle. segments, rays, or lines that — appear to be tangent to b. AB is a diameter because it is a chord that contains the center C. a circle are tangents. c. DE៮៮៮⃗ is a tangent ray because it is contained in a line that intersects the circle in exactly one point. d. ⃖AE៮៮⃗ is a secant because it is a line that intersects the circle in two points. MMonitoringonitoring PProgressrogress Help in English and Spanish at BigIdeasMath.com — — 1. In Example 1, what word best describes AG ? CB ? 2. In Example 1, name a tangent and a tangent segment. 530 Chapter 10 Circles hhs_geo_pe_1001.indds_geo_pe_1001.indd 530530 11/19/15/19/15 22:31:31 PMPM Drawing and Identifying Common Tangents CCoreore CConceptoncept Coplanar Circles and Common Tangents In a plane, two circles can intersect in two points, one point, or no points. Coplanar circles that intersect in one point are called tangent circles. Coplanar circles that have a common center are called concentric circles. no points of 1 point of intersection 2 points of intersection (tangent circles) intersection concentric circles A line or segment that is tangent to two coplanar circles is called a common tangent. A common internal tangent intersects the segment that joins the centers of the two circles. A common external tangent does not intersect the segment that joins the centers of the two circles. Drawing and Identifying Common Tangents Tell how many common tangents the circles have and draw them. Use blue to indicate common external tangents and red to indicate common internal tangents. a. b. c. SOLUTION Draw the segment that joins the centers of the two circles. Then draw the common tangents. Use blue to indicate lines that do not intersect the segment joining the centers and red to indicate lines that intersect the segment joining the centers. a. 4 common tangents b. 3 common tangents c. 2 common tangents MMonitoringonitoring PProgressrogress Help in English and Spanish at BigIdeasMath.com Tell how many common tangents the circles have and draw them. State whether the tangents are external tangents or internal tangents. 3. 4. 5. Section 10.1 Lines and Segments That Intersect Circles 531 hhs_geo_pe_1001.indds_geo_pe_1001.indd 531531 11/19/15/19/15 22:31:31 PMPM Using Properties of Tangents TTheoremsheorems Theorem 10.1 Tangent Line to Circle Theorem In a plane, a line is tangent to a circle if and only if the line is perpendicular to a radius of the circle at P its endpoint on the circle. Q m Line m is tangent to ⊙Q Proof Ex. 47, p. 536 if and only if m ⊥ QP. Theorem 10.2 External Tangent Congruence Theorem Tangent segments from a common external point R are congruent. S P T If SR and ST are tangent Proof Ex. 46, p. 536 segments, then SR ≅ ST. Verifying a Tangent to a Circle — Is ST tangent to ⊙P? T 35 12 S P 37 SOLUTION 2 + 2 = 2 Use the Converse of the Pythagorean— — Theorem— (Theorem 9.2). Because 12 35 37 , △PTS is a right triangle and ST ⊥ PT . So, ST is perpendicular to a radius of ⊙P at its endpoint on ⊙P. — By the Tangent Line to Circle Theorem, ST is tangent to ⊙P. Finding the Radius of a Circle In the diagram, point B is a point of tangency. Find the radius r of ⊙C. A 50 ft r C 80 ft r B SOLUTION — — You know from the Tangent Line to Circle Theorem that AB ⊥ BC , so △ABC is a right triangle. You can use the Pythagorean Theorem (Theorem 9.1). AC2 = BC2 + AB2 Pythagorean Theorem (r + 50)2 = r2 + 802 Substitute. r2 + 100r + 2500 = r2 + 6400 Multiply. 100r = 3900 Subtract r2 and 2500 from each side. r = 39 Divide each side by 100. The radius is 39 feet. 532 Chapter 10 Circles hhs_geo_pe_1001.indds_geo_pe_1001.indd 532532 11/19/15/19/15 22:31:31 PMPM Constructing a Tangent to a Circle Given ⊙C and point A, construct a line tangent to ⊙C that passes through A. Use a compass and straightedge. C A SOLUTION Step 1 Step 2 Step 3 C M A C M A C M A B B Find a— midpoint Draw a circle Construct a tangent line Draw AC . Construct the bisector Construct ⊙M with radius MA. Draw ⃖AB៮៮⃗ . It is a tangent of the segment and label the Label one of the points where to ⊙C that passes through A. midpoint M. ⊙M intersects ⊙C as point B. Using Properties of Tangents — — RS is tangent to ⊙C at S, and RT is tangent to ⊙C at T. Find the value of x. S 28 R C 3x + 4 T SOLUTION RS = RT External Tangent Congruence Theorem 28 = 3x + 4 Substitute. 8 = x Solve for x. The value of x is 8. MMonitoringonitoring PProgressrogress Help in English and Spanish at BigIdeasMath.com — — 6. Is DE tangent to ⊙C? 7. ST is tangent to ⊙Q. 8. Points M and N are Find the radius of ⊙Q. points of tangency. Find the value(s) of x. D M 3 Q x2 4 r P C r 18 2 E S 24 T N 9 Section 10.1 Lines and Segments That Intersect Circles 533 hhs_geo_pe_1001.indds_geo_pe_1001.indd 533533 11/19/15/19/15 22:31:31 PMPM 10.1 Exercises Dynamic Solutions available at BigIdeasMath.com VVocabularyocabulary andand CoreCore ConceptConcept CheckCheck 1. WRITING How are chords and secants alike? How are they different? 2. WRITING Explain how you can determine from the context whether the words radius and diameter are referring to segments or lengths. 3. COMPLETE THE SENTENCE Coplanar circles that have a common center are called _______. 4. WHICH ONE DOESN’T BELONG? Which segment does not belong with the other three? Explain your reasoning. chord radius tangent diameter MMonitoringonitoring ProgressProgress andand ModelingModeling withwith MathematicsMathematics In Exercises 5–10, use the diagram.